There is a simple path integral interpretation for (REF ): it is computed by a Euclidean path integral on an \(S^1 \times S^3\)  topology with complex structure parameters \(\omega _1,\omega _2\)  and periodic boundary conditions for the fermion fields around \(S^1\)  [1]}, [2]}, [3]}. The precise way the field theory is coupled to the background is determined by new-minimal background supergravity [4]}, [5]}. The metric on \(S^1 \times S^3\)  does not matter – it turns out to be \(\mathcal {Q}\) -exact (as long as it is Hermitian) [1]}, [2]}, [3]}. Due to this metric-independence, we can view this construction as a holomorphic twist.There is potentially a certain holomorphic anomaly [9]}, and see also [10]}, [11]}, [12]}, [13]}, [14]}. This is expected to (perhaps) affect the partition function only at order \(O(\beta )\)  in the high-temperature expansion and hence we do not expect that it will affect this paper, apart possibly for the \(O(\beta )\)  term.
